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arXiv · 1506.00467

Ehrhart polynomials with negative coefficients

Takayuki Hibi·Akihiro Higashitani·Akiyoshi Tsuchiya·Koutarou Yoshida
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Abstract

It is shown that, for each $d \geq 4$, there exists an integral convex polytope $\mathcal{P}$ of dimension $d$ such that each of the coefficients of $n, n^{2}, \ldots, n^{d-2}$ of its Ehrhart polynomial $i(\mathcal{P},n)$ is negative. Moreover, it is also shown that for each $d \geq 3$ and $1 \leq k \leq d-2$, there exists an integral convex polytope $\mathcal{P}$ of dimension $d$ such that the coefficient of $n^k$ of the Ehrhart polynomial $i(\mathcal{P},n)$ of $\mathcal{P}$ is negative and all its remaining coefficients are positive. Finally, we consider all the possible sign patterns of the coefficients of the Ehrhart polynomials of low dimensional integral convex polytopes.

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Takayuki Hibi, Akihiro Higashitani, Akiyoshi Tsuchiya, Koutarou Yoshida. 2015-06-01. Ehrhart polynomials with negative coefficients. https://arxiv.org/abs/1506.00467

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